197 research outputs found
Quantum walk approach to search on fractal structures
We study continuous-time quantum walks mimicking the quantum search based on
Grover's procedure. This allows us to consider structures, that is, databases,
with arbitrary topological arrangements of their entries. We show that the
topological structure of the database plays a crucial role by analyzing, both
analytically and numerically, the transition from the ground to the first
excited state of the Hamiltonian associated with different (fractal)
structures. Additionally, we use the probability of successfully finding a
specific target as another indicator of the importance of the topological
structure.Comment: 15 pages, 14 figure
Dynamics of continuous-time quantum walks in restricted geometries
We study quantum transport on finite discrete structures and we model the
process by means of continuous-time quantum walks. A direct and effective
comparison between quantum and classical walks can be attained based on the
average displacement of the walker as a function of time. Indeed, a fast growth
of the average displacement can be advantageously exploited to build up
efficient search algorithms. By means of analytical and numerical
investigations, we show that the finiteness and the inhomogeneity of the
substrate jointly weaken the quantum walk performance. We further highlight the
interplay between the quantum-walk dynamics and the underlying topology by
studying the temporal evolution of the transfer probability distribution and
the lower bound of long time averages.Comment: 25 pages, 13 figure
Quantum transport on two-dimensional regular graphs
We study the quantum-mechanical transport on two-dimensional graphs by means
of continuous-time quantum walks and analyse the effect of different boundary
conditions (BCs). For periodic BCs in both directions, i.e., for tori, the
problem can be treated in a large measure analytically. Some of these results
carry over to graphs which obey open boundary conditions (OBCs), such as
cylinders or rectangles. Under OBCs the long time transition probabilities
(LPs) also display asymmetries for certain graphs, as a function of their
particular sizes. Interestingly, these effects do not show up in the marginal
distributions, obtained by summing the LPs along one direction.Comment: 22 pages, 11 figure, acceted for publication in J.Phys.
Exactly solvable model of A + A \to 0 reactions on a heterogeneous catalytic chain
We present an exact solution describing equilibrium properties of the
catalytically-activated A + A \to 0 reaction taking place on a one-dimensional
lattice, where some of the sites possess special "catalytic" properties. The A
particles undergo continuous exchanges with the vapor phase; two neighboring
adsorbed As react when at least one of them resides on a catalytic site (CS).
We consider three situations for the CS distribution: regular, annealed random
and quenched random. For all three CS distribution types, we derive exact
results for the disorder-averaged pressure and present exact asymptotic
expressions for the particles' mean density. The model studied here furnishes
another example of a 1D Ising-type system with random multi-site interactions
which admits an exact solution.Comment: 7 pages, 3 Figures, appearing in Europhysics Letter
Survival probability of a particle in a sea of mobile traps: A tale of tails
We study the long-time tails of the survival probability of an
particle diffusing in -dimensional media in the presence of a concentration
of traps that move sub-diffusively, such that the mean square
displacement of each trap grows as with .
Starting from a continuous time random walk (CTRW) description of the motion of
the particle and of the traps, we derive lower and upper bounds for and
show that for these bounds coincide asymptotically, thus
determining asymptotically exact results. The asymptotic decay law in this
regime is exactly that obtained for immobile traps. This means that for
sufficiently subdiffusive traps, the moving particle sees the traps as
essentially immobile, and Lifshitz or trapping tails remain unchanged. For
and the upper and lower bounds again coincide,
leading to a decay law equal to that of a stationary particle. Thus, in this
regime the moving traps see the particle as essentially immobile. For ,
however, the upper and lower bounds in this regime no longer coincide
and the decay law for the survival probability of the particle remains
ambiguous
Lattice theory of trapping reactions with mobile species
We present a stochastic lattice theory describing the kinetic behavior of
trapping reactions , in which both the and particles
perform an independent stochastic motion on a regular hypercubic lattice. Upon
an encounter of an particle with any of the particles, is
annihilated with a finite probability; finite reaction rate is taken into
account by introducing a set of two-state random variables - "gates", imposed
on each particle, such that an open (closed) gate corresponds to a reactive
(passive) state. We evaluate here a formal expression describing the time
evolution of the particle survival probability, which generalizes our
previous results. We prove that for quite a general class of random motion of
the species involved in the reaction process, for infinite or finite number of
traps, and for any time , the particle survival probability is always
larger in case when stays immobile, than in situations when it moves.Comment: 12 pages, appearing in PR
Pascal Principle for Diffusion-Controlled Trapping Reactions
"All misfortune of man comes from the fact that he does not stay peacefully
in his room", has once asserted Blaise Pascal. In the present paper we evoke
this statement as the "Pascal principle" in regard to the problem of survival
of an "A" particle, which performs a lattice random walk in presence of a
concentration of randomly moving traps "B", and gets annihilated upon
encounters with any of them. We prove here that at sufficiently large times for
both perfect and imperfect trapping reactions, for arbitrary spatial dimension
"d" and for a rather general class of random walks, the "A" particle survival
probability is less than or equal to the survival probability of an immobile
target in the presence of randomly moving traps.Comment: 4 pages, RevTex, appearing in PR
Self-similar motion for modeling anomalous diffusion and nonextensive statistical distributions
We introduce a new universality class of one-dimensional iteration model
giving rise to self-similar motion, in which the Feigenbaum constants are
generalized as self-similar rates and can be predetermined. The curves of the
mean-square displacement versus time generated here show that the motion is a
kind of anomalous diffusion with the diffusion coefficient depending on the
self-similar rates. In addition, it is found that the distribution of
displacement agrees to a reliable precision with the q-Gaussian type
distribution in some cases and bimodal distribution in some other cases. The
results obtained show that the self-similar motion may be used to describe the
anomalous diffusion and nonextensive statistical distributions.Comment: 15pages, 5figure
Kinetics of stochastically-gated diffusion-limited reactions and geometry of random walk trajectories
In this paper we study the kinetics of diffusion-limited, pseudo-first-order
A + B -> B reactions in situations in which the particles' intrinsic
reactivities vary randomly in time. That is, we suppose that the particles are
bearing "gates" which interchange randomly and independently of each other
between two states - an active state, when the reaction may take place, and a
blocked state, when the reaction is completly inhibited. We consider four
different models, such that the A particle can be either mobile or immobile,
gated or ungated, as well as ungated or gated B particles can be fixed at
random positions or move randomly. All models are formulated on a
-dimensional regular lattice and we suppose that the mobile species perform
independent, homogeneous, discrete-time lattice random walks. The model
involving a single, immobile, ungated target A and a concentration of mobile,
gated B particles is solved exactly. For the remaining three models we
determine exactly, in form of rigorous lower and upper bounds, the large-N
asymptotical behavior of the A particle survival probability. We also realize
that for all four models studied here such a probalibity can be interpreted as
the moment generating function of some functionals of random walk trajectories,
such as, e.g., the number of self-intersections, the number of sites visited
exactly a given number of times, "residence time" on a random array of lattice
sites and etc. Our results thus apply to the asymptotical behavior of the
corresponding generating functions which has not been known as yet.Comment: Latex, 45 pages, 5 ps-figures, submitted to PR
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